A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.